Aharonov--Bohm effect, flat connections, and Green's theorem
The Aharonov-Bohm (A-B) effect has been in the centre of fame in a major part of theoretical physics during the past four decades. It is, indeed, one of the simplest and most vital examples of the interplay between classical gauge theory, quantum mechanics, differential geometry and topology. Here the authors have furnished to begin with a modern proof of the generalized Green's and Stokes' theorems followed by a fiber-bundle interpretation of the A-B effect in section 3. The next section is devoted at length to the investigation of the general study of connections on a product bundle, and in particular of the subspace of flat connections in the case of a connected abelian structural group \(G\). In section 5, the holonomy group of each equivalence class of flat connection on \(MXU(1) @>\pi>>M\), where \(M=R^2-\{0\}\), is calculated. The results of section 4 are particularized to the A-B connection and finally in section 6, certain remarks and comments on the appearance of the A-B field \(A=({\Phi\over 2\pi}) {xdy-ydx \over x^2+y^2}\) in a vortex solution to the abelian Higgs model are given. There are 15 references, 4 theorems, 8 lemmas, 3 propositions and 4 corollaries with extensive proofs, in this 22-page contribution.
scientific article; zbMATH DE number 1765088